Siegel modular form
E871397
A Siegel modular form is a type of complex analytic function defined on the Siegel upper half-space that generalizes classical modular forms to higher dimensions and plays a central role in number theory and algebraic geometry.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Siegel modular form canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T10543859 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Siegel modular form Context triple: [Carl Ludwig Siegel, notableWork, Siegel modular form]
-
A.
modular j-invariant
The modular j-invariant is a fundamental modular function that classifies complex elliptic curves up to isomorphism and plays a central role in number theory, complex analysis, and the theory of modular forms.
-
B.
Hecke operators
Hecke operators are algebraic operators acting on modular forms that play a central role in number theory, particularly in understanding congruences, L-functions, and the arithmetic of modular forms.
-
C.
Selberg trace formula
The Selberg trace formula is a fundamental result in analytic number theory and spectral theory that relates lengths of closed geodesics on a Riemannian manifold to the spectrum of its Laplace operator, serving as a non-abelian analogue of the Poisson summation formula.
-
D.
Hecke theory
Hecke theory is a branch of number theory centered on Hecke operators and modular forms, providing powerful tools to study arithmetic properties of modular forms and related objects.
-
E.
Shimura varieties
Shimura varieties are higher-dimensional algebraic varieties that generalize modular curves and play a central role in the Langlands program by connecting number theory, automorphic forms, and arithmetic geometry.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Siegel modular form Target entity description: A Siegel modular form is a type of complex analytic function defined on the Siegel upper half-space that generalizes classical modular forms to higher dimensions and plays a central role in number theory and algebraic geometry.
-
A.
modular j-invariant
The modular j-invariant is a fundamental modular function that classifies complex elliptic curves up to isomorphism and plays a central role in number theory, complex analysis, and the theory of modular forms.
-
B.
Hecke operators
Hecke operators are algebraic operators acting on modular forms that play a central role in number theory, particularly in understanding congruences, L-functions, and the arithmetic of modular forms.
-
C.
Selberg trace formula
The Selberg trace formula is a fundamental result in analytic number theory and spectral theory that relates lengths of closed geodesics on a Riemannian manifold to the spectrum of its Laplace operator, serving as a non-abelian analogue of the Poisson summation formula.
-
D.
Hecke theory
Hecke theory is a branch of number theory centered on Hecke operators and modular forms, providing powerful tools to study arithmetic properties of modular forms and related objects.
-
E.
Shimura varieties
Shimura varieties are higher-dimensional algebraic varieties that generalize modular curves and play a central role in the Langlands program by connecting number theory, automorphic forms, and arithmetic geometry.
- F. None of above. chosen
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
automorphic form
ⓘ
complex analytic function ⓘ mathematical object ⓘ |
| appearsIn | theory of Siegel modular threefolds ⓘ |
| canBeInterpretedAs | section of line bundle on Siegel modular variety ⓘ |
| cuspFormCondition | vanishing at all cusps ⓘ |
| definedOn | Siegel upper half-space ⓘ |
| degree | positive integer g ⓘ |
| degree1SpecialCase | elliptic modular form ⓘ |
| domain | space of symmetric complex matrices with positive definite imaginary part ⓘ |
| field |
algebraic geometry
ⓘ
number theory ⓘ |
| FourierCoefficientsIndexedBy | symmetric, semi-positive definite integral matrices ⓘ |
| generalizes |
Eisenstein series
ⓘ
classical modular form ⓘ cusp forms ⓘ |
| group |
symplectic group Sp(2g,ℤ)
ⓘ
linked to:
symplectic group Sp(2n,R)
|
| groupAction | fractional linear transformation on Siegel upper half-space ⓘ |
| hasCongruenceSubgroupVersion | Siegel modular form for congruence subgroup of Sp(2g,ℤ) ⓘ |
| hasFourierExpansion | yes ⓘ |
| hasHeckeAction | Hecke operators for Sp(2g,ℤ) ⓘ |
| hasParameter |
character
ⓘ
degree ⓘ level ⓘ weight ⓘ |
| hasStructure | graded ring by weight ⓘ |
| hasVectorValuedVersion | vector-valued Siegel modular form ⓘ |
| namedAfter | Carl Ludwig Siegel ⓘ |
| playsCentralRoleIn |
Langlands program
ⓘ
higher-dimensional modular form theory ⓘ |
| relatedTo |
Jacobi form
ⓘ
Siegel modular variety ⓘ
linked to:
Siegel modular varieties
Siegel upper half-space ⓘ automorphic representation ⓘ theta function ⓘ |
| specialCase |
Siegel Eisenstein series
ⓘ
linked to:
Eisenstein series
Siegel cusp form ⓘ |
| studiedBy |
Carl Ludwig Siegel
ⓘ
Goro Shimura ⓘ Jun-ichi Igusa ⓘ |
| transformationProperty | invariance under symplectic group action ⓘ |
| usedIn |
arithmetic geometry
ⓘ
moduli of principally polarized abelian varieties ⓘ theory of L-functions ⓘ theory of abelian varieties ⓘ |
| usedToConstruct |
Siegel modular L-function
ⓘ
linked to:
L-functions
|
| usedToDefine | Hecke eigenforms for symplectic groups ⓘ |
| weightCondition | holomorphic with polynomial growth at infinity ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Siegel modular form Description of subject: A Siegel modular form is a type of complex analytic function defined on the Siegel upper half-space that generalizes classical modular forms to higher dimensions and plays a central role in number theory and algebraic geometry.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.